Displaying similar documents to “ Gl n ( R ) -invariant variational principles on frame bundles.”

Invariant variational problems on principal bundles and conservation laws

Ján Brajerčík (2011)

Archivum Mathematicum

Similarity:

In this work, we consider variational problems defined by G -invariant Lagrangians on the r -jet prolongation of a principal bundle P , where G is the structure group of P . These problems can be also considered as defined on the associated bundle of the r -th order connections. The correspondence between the Euler-Lagrange equations for these variational problems and conservation laws is discussed.

Order reduction of the Euler-Lagrange equations of higher order invariant variational problems on frame bundles

Ján Brajerčík (2011)

Czechoslovak Mathematical Journal

Similarity:

Let μ : F X → X be a principal bundle of frames with the structure group Gl n ( ℝ ) . It is shown that the variational problem, defined by Gl n ( ℝ ) -invariant Lagrangian on J r F X , can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equations.

On a generalization of Helmholtz conditions

Radka Malíková (2009)

Acta Mathematica Universitatis Ostraviensis

Similarity:

Helmholtz conditions in the calculus of variations are necessary and sufficient conditions for a system of differential equations to be variational ‘as it stands’. It is known that this property geometrically means that the dynamical form representing the equations can be completed to a closed form. We study an analogous property for differential forms of degree 3, so-called Helmholtz-type forms in mechanics ( n = 1 ), and obtain a generalization of Helmholtz conditions to this case. ...

Geometric mechanics on nonholonomic submanifolds

Olga Krupková (2010)

Communications in Mathematics

Similarity:

In this survey article, nonholonomic mechanics is presented as a part of geometric mechanics. We follow a geometric setting where the constraint manifold is a submanifold in a jet bundle, and a nonholonomic system is modelled as an exterior differential system on the constraint manifold. The approach admits to apply coordinate independent methods, and is not limited to Lagrangian systems under linear constraints. The new methods apply to general (possibly nonconservative) mechanical...

A new Lagrangian dynamic reduction in field theory

François Gay-Balmaz, Tudor S. Ratiu (2010)

Annales de l’institut Fourier

Similarity:

For symmetric classical field theories on principal bundles there are two methods of symmetry reduction: covariant and dynamic. Assume that the classical field theory is given by a symmetric covariant Lagrangian density defined on the first jet bundle of a principal bundle. It is shown that covariant and dynamic reduction lead to equivalent equations of motion. This is achieved by constructing a new Lagrangian defined on an infinite dimensional space which turns out to be gauge group...